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Some Combinatorics Related to Central Binomial Coefficients: Grand-Dyck Paths, Coloured Noncrossing Partitions and Signed Pattern Avoiding Permutations

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Abstract

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.

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References

  1. Bandlow J., Killpatrick K.: An area-to-inv bijection between Dyck paths and 312-avoiding permutations. Electron. J. Combin. 8, 16 (2001) #R40

    MathSciNet  Google Scholar 

  2. Barcucci E., Bernini A., Ferrari L., Poneti M.: A distributive lattice structure connecting Dyck paths, noncrossing partitions and 312-avoiding permutations. Order 22, 311–328 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barcucci E., Del Lungo A., Pergola E., Pinzani R.: ECO: a methodology for the enumeration of combinatorial objects. J. Differ. Equ. Appl. 5, 435–490 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bergeron F., Labelle G., Leroux P.: Combinatorial species and tree-like structures. Encyclopedia of Mathematics and its Applications, vol. 67. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  5. Bergeron N., Zabrocki M.: The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree. J. Algebra Appl. 8, 581–600 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bernini A., Ferrari L.: Order properties of the Motzkin and Schröder families. Australasian J. Combin. 39, 259–272 (2007)

    MATH  MathSciNet  Google Scholar 

  7. Bernini A., Ferrari L., Pinzani R.: Enumerating permutations avoiding three Babson-Steingrímsson patterns. Ann. Combin. 9, 137–162 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Björner A., Brenti F.: Combinatorics of Coxeter groups. Graduate Text in Mathematics, vol. 231. Springer, New York (2005)

    Google Scholar 

  9. Deutsch E.: A bijection on Dyck paths and its consequences. Discrete Math. 179, 253–256 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ferrari L., Pinzani R.: Lattices of lattices paths. J. Statist. Plann. Inference 135, 77–92 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Klazar M.: Bell numbers, their relatives, and algebraic differential equations. J. Combin. Theory Ser. A 102, 63–87 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Krattenthaler C.: Permutations with restricted patterns and Dyck paths. Adv. Appl. Math. 27, 510–530 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Labelle G., Leroux P., Pergola E., Pinzani R.: Stirling numbers interpolation using permutations with forbidden subsequences. Discrete Math. 246, 177–195 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Narayana T.V., Fulton G.E.: A note on the compositions of an integer. Can. Math. Bull. 1, 169–173 (1958)

    MATH  MathSciNet  Google Scholar 

  15. Pergola E., Pinzani R., Rinaldi S.: Approximating algebraic functions by means of rational ones. Theor. Comput. Sci. 270, 643–657 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Reiner V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177, 195–222 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shapiro L.W., Getu S., Woan W.J., Woodson L.C.: The Riordan group. Discrete Appl. Math. 34, 229–239 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  18. Simion R.: Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations. Electron. J. Combin. 7, 27 (2000) #R9

    MathSciNet  Google Scholar 

  19. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, at http://www.research.att.com/~njas/sequences/index.html

  20. Stanton D.: Unimodality and Young’s lattice. J. Combin. Theory A 54, 41–53 (1990)

    Article  MATH  Google Scholar 

Download references

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  1. Dipartimento di Sistemi e Informatica, University of Firenze, viale Morgagni 65, 50134, Firenze, Italy

    Luca Ferrari

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  1. Luca Ferrari
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Correspondence to Luca Ferrari.

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Ferrari, L. Some Combinatorics Related to Central Binomial Coefficients: Grand-Dyck Paths, Coloured Noncrossing Partitions and Signed Pattern Avoiding Permutations. Graphs and Combinatorics 26, 51–70 (2010). https://doi.org/10.1007/s00373-010-0895-z

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  • DOI: https://doi.org/10.1007/s00373-010-0895-z

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